Theorem chdmm2

Description: De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	chdmm2	|- ( ( A e. CH /\ B e. CH ) -> ( _|_ ` ( ( _|_ ` A ) i^i B ) ) = ( A vH ( _|_ ` B ) ) )

Proof

Step	Hyp	Ref	Expression
1		choccl	|- ( A e. CH -> ( _|_ ` A ) e. CH )
2		chdmm1	|- ( ( ( _|_ ` A ) e. CH /\ B e. CH ) -> ( _|_ ` ( ( _|_ ` A ) i^i B ) ) = ( ( _|_ ` ( _|_ ` A ) ) vH ( _|_ ` B ) ) )
3	1 2	sylan	|- ( ( A e. CH /\ B e. CH ) -> ( _|_ ` ( ( _|_ ` A ) i^i B ) ) = ( ( _|_ ` ( _|_ ` A ) ) vH ( _|_ ` B ) ) )
4		ococ	|- ( A e. CH -> ( _|_ ` ( _|_ ` A ) ) = A )
5	4	adantr	|- ( ( A e. CH /\ B e. CH ) -> ( _|_ ` ( _|_ ` A ) ) = A )
6	5	oveq1d	|- ( ( A e. CH /\ B e. CH ) -> ( ( _|_ ` ( _|_ ` A ) ) vH ( _|_ ` B ) ) = ( A vH ( _|_ ` B ) ) )
7	3 6	eqtrd	|- ( ( A e. CH /\ B e. CH ) -> ( _|_ ` ( ( _|_ ` A ) i^i B ) ) = ( A vH ( _|_ ` B ) ) )